For questions regarding harmonic functions.

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The harmonic functions are smooth

Let $u \in C^2(\Omega)$ a harmonic function, then $u \in C^{\infty}(\Omega)$. Here $\Omega$ is a domain of $\mathbb{R}^{n}$. Step 1 for proof: $u \equiv u^{\epsilon}$ in $\Omega_{\epsilon}$, for $\...
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135 views

About a harmonic function in the upper half plane [duplicate]

I'm struggling with the following question: Suppose that $C$ is a positive constant, $u$ is harmonic in the upper half plane $\mathrm{Im}z>0$, and that $0 \le u(z) \le C\mathrm{Im}z$ for $\mathrm{...
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Optimizing value of discrete harmonic function at a given point

Let $n>0$, and let $S_n$ denote the discrete square $S_n=[|-n,n|]^2$ (so $S_n$ has $(2n+1)^2$ elements). Let $K_n$ denote the set of four corner points $\lbrace (\pm n,\pm n)\rbrace$, and $C_n=S_n\...
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27 views

Mean Value Property - Nonnegative Harmonic function

I want to prove that the mean value property $$u(\textbf{x}_0) = \frac{1}{\pi r^2} \int \int _{\left \{ \left | x_0-x \right < r| \right \}} u(\textbf{x})d\textbf{x}$$ for non-negative harmonic ...
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Fourier series: can a function be odd and have a dc component?

Long story short: fourier series is taken in two subjects (for now). One doc says that the dc component is 0 if the function is odd. The other says that odd and even has no effect on the dc ...
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47 views

Equivalent definition of harmonic functions

Let $\Omega \subset \mathbb{R}^n$ be a domain. And let $u \in \mathcal{C}(\Omega)$. Prove that: The function u is harmonic in $\Omega$, i.e. $u \in \mathcal{C}^2(\Omega)$ and $\Delta u = 0$ on $\...
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39 views

Uniform Convergence: Poisson Kernel

If we fix $θ_∗ > 0$, then $P(r, θ) → 0$ uniformly on the set $ \left\lbrace θ : |θ| ≥ θ_∗ \right\rbrace $ as $r → a^-$ $$P(r,\theta) = \frac{a^2-r^2}{a^2-2r\cos(\theta)+r^2}$$ $0\leq r <a,\...
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1answer
31 views

Plotting Spherical Coordinates

I'm trying to plot the Poisson Kernel, where a = 1, so the resulting equation would be $$P(r,\theta) = \frac{1-r^2}{1-2r\cos(\theta)+r^2}$$ $0\leq r <a = 1 ,\textrm{ } -\pi \leq \theta \leq \pi$ ...
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Harmonicity on semisimple groups

Let $G$ be a semisimple real Lie group, $U(\mathfrak{g})$ its universal enveloping algebra, let $\Omega$ be the Casimir element in $U(\mathfrak{g})$ and let $f$ be a smooth (or analytic) real-valued ...
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1answer
24 views

Bounds on derivatives of harmonic functions on unit ball

Let $u$ be a harmonic function on the unit ball in $\mathbb{R}^n$. Show that $$\sup_{B_{1/2}} \lvert \nabla u \rvert \leq C(n) \sup_{\partial B_1} \lvert u \rvert$$ More generally, show that $$\...
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Mean value property for harmonic functions (geometric idea)

Someone knows a bibliographic reference geometrically explore the idea of ​​the mean property for harmonic functions in domains of $\mathbb{R}^{n}$ ?
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2answers
39 views

Newton potential for Neumann problem on unit disk

Problem: Show that $$G(x, y) = \frac{1}{2\pi} \left( \log \lvert x - y \rvert + \log \left\lvert \frac{x}{\lvert x \rvert} - \lvert x \rvert y \right\rvert \right)$$ is a Green's function for the ...
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Find the complex potential given the real part $\frac{6000}{\pi}\theta$

In my 'Complex Variable Theory' notes, we have an example where we use a linear fractional mapping to solve for Laplace's equation between semi-circular plates. The Question reads: Find the ...
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2answers
168 views

How to prove Liouville's theorem for subharmonic functions

I noticed this post and this paper, which gives a version of Liouville's theorem for subharmonic functions and the reference of its proof, but I think there must be an easier proof for the following ...
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1answer
15 views

Are there identities which will make calculating $\Delta \frac{e^{ik|x|}}{|x|}$ more efficient and clearer?

Let $x = (x_1, x_2, x_3)$ be a point in $\mathbb{R}^3$. I am trying to calculate $$\Delta \frac{e^{ik|x|}}{|x|}$$ Doing this 'manually' be calculating second derivatives is really long and tedious ...
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Laplace transform

Hello how can I solve this problem $ty''-y'=t^2$, $y (0)=0 $ I though about several ways like $y''-y'/t =t$ but I did not know how to solve the integral of $( y'/t) $. Then I thought about ...
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Stuck while trying to show fundamental solution $\frac{1}{2\pi}\ln|x|$ satisfies $\Delta \phi(x) = \delta(x)$ in 2d?

I am trying to show that the fundamental solution to the Laplacian in 2D satisfies $$\Delta \phi(x) = \delta(x)$$ where $x = (x_1, x_2) \in \mathbb{R}^2$. So the fundamental solution in 2D is $\...
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1answer
36 views

Properties of harmonic function on $\mathbb{R}^2$

Assume $f$ is harmonic on $\mathbb{R}^2$. I want to prove that if there exists a constant $M$ such that $f(x,y) \geq M$ for all $(x,y)\in \mathbb{R}^2$, then $f$ must be a constant fuction. I'm ...
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Computing $\int_D x^3-3xy^2\,{\rm d}x\,{\rm d}y$ using the mean value property.

I am asked to compute $$\int_D x^3-3xy^2\,{\rm d}x\,{\rm d}y,$$where $D = \{ (x,y) \mid (x+1)^2+y^2 \leq 9, \text{and }(x-1)^2+y^2 \geq 1 \}$. Granted, $u(x,y) = x^3-3xy^2$ is harmonic (it is the real ...
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30 views

What does it mean for a complex-valued function to be bounded above (or below)?

I was reading about the maximum-minimum principle for harmonic functions in my lecture notes, and it was formulated like this: Let $\phi$ be harmonic in a simply-connected domain $D$. If $\phi$ is ...
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1answer
28 views

Relation between Poisson kernel and harmonic measure

If $D$ is a domain in the complex plane bounded by a Jordan curve $J$, what's the relation between the harmonic measure and the Poisson kernel on the boundary? More specifically, if $z_0 \in D$ and $...
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1answer
118 views

Harmonic function with injective boundary conditions is an immersion?

This question is in some sense a continuation of this question, though more focused in its scope. Let $(M,g)$ be an $n$-dimensional, connected, compact Riemannian manifold with boundary. Assume we ...
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1answer
45 views

Does harmonic decomposition preserve immersions?

In a nutshell: If we start from an immersion, and look at its harmonic decomposition, are the components also immersions? Details: Let $(M,g)$ be an $n$-dimensional, compact Riemannian manifold with ...
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1answer
33 views

The value of a harmonic function in the interior of a unit disk

Let $u(z)$ be a bounded harmonic function in $D$ such that the limit $$\lim_{r→1^-}u(re^{iφ})$$ is equal to 1 when for $0 < φ < π$ and to 0 for $π < φ < 2π$. Find $u(1/2)$. To ...
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1answer
61 views

Separation of variables for PDE: dividing by zero?

This feels like a question that is both simple and duplicate but I can't find an answer or a previous version of the question. Suppose we are given some PDE, for example Laplace's Equation in polar ...
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1answer
20 views

Use result that composition of analytic functions is harmonic to find harmonic conjugate of $e^{-2xy}\sin (x^{2}-y^{2})$

Using this result, I need to find a harmonic conjugate for $e^{-2xy}\sin(x^{2}-y^{2})$. In that result, I'm supposing that $s = e^{-2xy}$ and $t = \sin(x^{2}-y^{2})$, but I really don't know how to ...
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3 dimensional harmonic conjugates?

An $n$ dimensional harmonic function is defined to be a real valued function $f$ in $\mathbb{R}^n$ such that $\nabla^2 f = 0 $. Equivalently, $f$ is the scalar potential of a conservative vector field ...
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95 views

Show composition of harmonic function and analytic function is harmonic without calculating 2nd derivatives

This is not a duplicate. I realize similar questions to this have been asked, but what I am asking is slightly different: I need to prove the following, arguing by complex differentiability only, and ...
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Critical points of a harmonic function

Suppose $\phi$ is harmonic on some compact, connected region of $\mathbb{R}^3$. Is there an algorithm that is guaranteed to find all critical points of $\phi$? (Obviously, these will all be saddle ...
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1answer
40 views

Find a harmonic conjugate for the function $u(x,y)=x^{3}+Axy^{2}$ if one exists - $u(x,y)$ might not be harmonic?

For $u(x,y) = x^{3} + Axy^{2}$, where $A$ is a real number, I need to find a harmonic conjugate, or if one does not exist, show that it does not exist. I began my approach to this problem by first ...
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Square integrable functions on the unit ball

In one dimension, the space $L^{2}([0,1], dx)$ of complex-valued square integrable functions on $[0,1]$ is well known to be separable, and sine and cosine provide an explicit Hilbert basis for it. My ...
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1answer
34 views

Find a harmonic function which goes to $0$ on the boundary, which is not identically $0$

Find a function which is harmonic on the area bounded by positive x axis and the line $y=x$, which goes to $0$ on the boundary, which is not identically $0$. Why doesn't it violate the max/min ...
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1answer
43 views

Intuition behind $\nabla \cdot \frac{1}{\rho}\nabla$?

Oftentimes instead of the Laplacian I notice the very similar operator $$\nabla \cdot \frac{1}{\rho}\nabla$$ What is the intuition behind this operator? How does it differ intuitively from the ...
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69 views

Fourier Transform to solve Laplace's equation in cylindrical coordinates

I am trying to solve $\nabla^2 u = 0$ in cylindrical polar coordinates (and radial symmetry) $$ \frac{\partial^2 u}{\partial r^2} + \frac{1}{r} \frac{\partial u}{\partial r} + \frac{\partial^2 u}{\...
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1answer
13 views

Laplace's equation 2 variable PDE/chain rule show function is a solution

The question is: 'Show that if f(x,y) is harmonic, then $f(x^2-y^2,2xy)$ is also harmonic using Laplace's equation: $\frac{\partial^2f}{\partial x^2} + \frac{\partial^2f}{\partial y^2} = 0 $. I end ...
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1answer
108 views

Find Green's function of quarter-plane with method of images

Consider a domain $$D= \{(x,y) : x>0, y>0 \}$$ Let $\textbf x = (x,y)$ and $\xi =(\xi_x , \xi_y)$. Find the Green's function, $G(\textbf x , \xi)$ such that $$\nabla ^2 G=\delta (\textbf x - \xi)...
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1answer
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Inequality involving harmonic functions over the ball and half ball

Let $B\subset \mathbb R^2$ be a unit ball. Let $B^+:=B\cap \{x_2\geq 0\}$ where we set $x=(x_1,x_2)\in \mathbb R^2$. Let $\omega\in C^1(\partial B)$ be given such that $|\nabla \omega|>0$ for all $...
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31 views

Verify that Poisson's integral formula is harmonic on the unit disk

Let $D = \{ z : |z| < 1\}$ be the unit disk and let $z \in D$ and $e^{i\theta} \in \partial D$. Let $h(e^{i\theta})$ be a function defined on $\partial D$ and let $\tilde h(z)$ be defined in $D$. ...
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1answer
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Greens function for 2d laplace equation with neumann boundary conditions

I have a domain, $ D : {(x,y) : x>0 , y>0}$ Let $ \mathbf{x}= (x,y) $ and $\mathbf{\xi}= (\xi_x, \xi_y)$, The Greens function satisfying: $$\nabla^2G = \delta(\mathbf{x} - \mathbf{\xi} )$$ ...
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Is it posible to solve the Laplce equation on an open set with dirac delta boundary conditions?

By dirac deta boundary conditions, i mean there is an $y \in \partial D$ such that the value on that point is $\infty$ and 0 elsewhere. Intuitivelly i would think it should be true: the Laplace ...
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Show that $z^n+c$ is harmonic, $c\in\mathbb C$.

I would like to know how to show $$f(z) = z^n+c$$ for $c\in \mathbb C$ is harmonic over $D =\{|z|\leq r\}$. I know that if I express $z = x+iy$, then I can have $f=u+iv$, where $u$ and $v$ will be ...
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Intersections of the level curves of two (conjugate) harmonic functions

Let $f:\mathbb{R}^2\rightarrow\mathbb{R}$ with $f(x,y)=e^{-x}(x\sin y-y\cos y)$. 1 Let $g$ be one of the conjugate harmonics of $f$ on $\mathbb{R}^2$ and assume the level curves of $f$ and $g$ ...
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What is the harmonic conjugate of $u=4xy-3x+5y$?

What is the harmonic conjugate of $u=4xy-3x+5y$? I got $u'x=4-y=v'y$ then I integrated $v'y$ to get $v= 2y^2-3y+h(x)$. Then I did $-u'y=v'x$ so, $5= h'(x)$ then I integrated $5$ with respect to $x$ ...
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Why is $\int_{B(0,\epsilon)}|\Phi(y)|dy$ bounded?

Why is $\int_{B(0,\epsilon)}|\Phi(y)|dy$ bounded ? If $\Phi$ is the fundamental solution of Laplace equation defined by $\Phi(x)=\begin{cases}-\frac1{2\pi}\log|x|& \text{if n=2}\\\frac{1}{n(n-2)\...
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22 views

harmonic measure circle

I'm trying to compare the probability of a particle (performing Brownian Motion), starting a large distance away from a circle, passing through a specific section of that circle is approximately the ...
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1answer
50 views

Mean value of a subharmonic function, divided by the logarithm of radius, has a limit

I am pretty stuck on a homework problem on harmonic functions, or rather subharmonic functions (which for us are allowed to take the value $-\infty$). The statement is as follows: Supper $u$ is ...
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Deduce the definition of a harmonic function in the context of a Markov Chain

We know from PDE that a harmonic function $f$ satisfies the mean value property, namely, $f(x)$ = $\frac{1}{\vert{B_r(x)}\vert}\int_{B_r(x)}f(y)dy$ where $B_r(x)$ is the ball about $x$ with radius $r$....
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1answer
90 views

A harmonic function with sublinear growth at infinity is constant

Show that if $u$ is harmonic in $\mathbb R^n$ and $u=o(|x|)$, then $u$ is a constant.(Hint: use the solid version of the mean value property $u(x)=\frac{1}{\omega R^n} \int _{B_{R(x)}} u(y)dy$, and ...
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24 views

biharmonic complex analysis

Complex analysis texts typically discuss analytic functions whose real and imaginary components are harmonic and satisfy the Laplace equation, $\nabla^2 f = 0$. I am working with a complex function ...
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let $f(x)$ is real polynomial. Can we say that $f(\left| x \right|)$ is subharmonic?

A continuous function $\varphi :\mathbb{R} \to \mathbb{C}$ is subharmonic if and only if, for any closed disc in $U$ with centre $\lambda_0$ and radius $r$,$$\varphi ({\lambda _0}) \le \frac{1}{{2\pi }...